# Classical modular forms downloaded from the LMFDB on 27 July 2026. # Search link: https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/1472/ # Query "{'level': 1472}" returned 109 forms, sorted by analytic conductor. # Each entry in the following data list has the form: # [Label, Dim, $A$, Field, CM, RM, Traces, Fricke sign, $q$-expansion] # For more details, see the definitions at the bottom of the file. "1472.1.f.a" 1 0.7346236985957796 "1.1.1.1" [-23] [] [0, -1, 0, 0] NULL "q-q^{3}+q^{13}-q^{23}+q^{25}+q^{27}+\\cdots" "1472.1.f.b" 1 0.7346236985957796 "1.1.1.1" [-23] [] [0, 1, 0, 0] NULL "q+q^{3}+q^{13}+q^{23}+q^{25}-q^{27}+\\cdots" "1472.1.k.a" 2 0.7346236985957796 "2.0.4.1" [-23] [] [0, -2, 0, 0] NULL "q+(-i-1)q^{3}+i q^{9}+(-i-1)q^{13}+\\cdots" "1472.1.k.b" 4 0.7346236985957796 "4.0.144.1" [-23] [] [0, 2, 0, 0] NULL "q+(-\\zeta_{12}+\\zeta_{12}^{2})q^{3}+(\\zeta_{12}^{2}-\\zeta_{12}^{3}+\\cdots)q^{9}+\\cdots" "1472.1.s.a" 8 0.7346236985957796 "8.0.16777216.1" [-23] [] [0, 0, 0, 0] NULL "q+\\zeta_{16}q^{2}+(\\zeta_{16}-\\zeta_{16}^{4})q^{3}+\\zeta_{16}^{2}q^{4}+\\cdots" "1472.1.s.b" 16 0.7346236985957796 "16.0.1846757322198614016.1" [-23] [] [0, 0, 0, 0] NULL "q+\\zeta_{48}q^{2}+(-\\zeta_{48}^{4}+\\zeta_{48}^{17})q^{3}+\\cdots" "1472.2.a.a" 1 11.753979177532473 "1.1.1.1" [] [] [0, -3, 0, -2] -1 "q-3q^{3}-2q^{7}+6q^{9}+5q^{13}-6q^{17}+\\cdots" "1472.2.a.b" 1 11.753979177532473 "1.1.1.1" [] [] [0, -3, 2, 4] -1 "q-3q^{3}+2q^{5}+4q^{7}+6q^{9}+2q^{11}+\\cdots" "1472.2.a.c" 1 11.753979177532473 "1.1.1.1" [] [] [0, -1, 0, 2] 1 "q-q^{3}+2q^{7}-2q^{9}+q^{13}-6q^{17}+\\cdots" "1472.2.a.d" 1 11.753979177532473 "1.1.1.1" [] [] [0, -1, 2, 4] 1 "q-q^{3}+2q^{5}+4q^{7}-2q^{9}-2q^{11}+\\cdots" "1472.2.a.e" 1 11.753979177532473 "1.1.1.1" [] [] [0, -1, 4, -2] -1 "q-q^{3}+4q^{5}-2q^{7}-2q^{9}-4q^{11}+\\cdots" "1472.2.a.f" 1 11.753979177532473 "1.1.1.1" [] [] [0, 0, -4, -4] -1 "q-4q^{5}-4q^{7}-3q^{9}-2q^{11}+2q^{13}+\\cdots" "1472.2.a.g" 1 11.753979177532473 "1.1.1.1" [] [] [0, 0, -4, 4] -1 "q-4q^{5}+4q^{7}-3q^{9}+2q^{11}+2q^{13}+\\cdots" "1472.2.a.h" 1 11.753979177532473 "1.1.1.1" [] [] [0, 0, 0, -4] -1 "q-4q^{7}-3q^{9}+6q^{11}+2q^{13}+6q^{17}+\\cdots" "1472.2.a.i" 1 11.753979177532473 "1.1.1.1" [] [] [0, 0, 0, 4] -1 "q+4q^{7}-3q^{9}-6q^{11}+2q^{13}+6q^{17}+\\cdots" "1472.2.a.j" 1 11.753979177532473 "1.1.1.1" [] [] [0, 1, 0, -2] 1 "q+q^{3}-2q^{7}-2q^{9}+q^{13}-6q^{17}+\\cdots" "1472.2.a.k" 1 11.753979177532473 "1.1.1.1" [] [] [0, 1, 2, -4] 1 "q+q^{3}+2q^{5}-4q^{7}-2q^{9}+2q^{11}+\\cdots" "1472.2.a.l" 1 11.753979177532473 "1.1.1.1" [] [] [0, 1, 4, 2] -1 "q+q^{3}+4q^{5}+2q^{7}-2q^{9}+4q^{11}+\\cdots" "1472.2.a.m" 1 11.753979177532473 "1.1.1.1" [] [] [0, 3, 0, 2] -1 "q+3q^{3}+2q^{7}+6q^{9}+5q^{13}-6q^{17}+\\cdots" "1472.2.a.n" 1 11.753979177532473 "1.1.1.1" [] [] [0, 3, 2, -4] -1 "q+3q^{3}+2q^{5}-4q^{7}+6q^{9}-2q^{11}+\\cdots" "1472.2.a.o" 2 11.753979177532473 "2.2.8.1" [] [] [0, -2, 4, -4] 1 "q+(-1+\\beta )q^{3}+(2-\\beta )q^{5}+(-2+\\beta )q^{7}+\\cdots" "1472.2.a.p" 2 11.753979177532473 "2.2.17.1" [] [] [0, -1, -4, 0] 1 "q-\\beta q^{3}-2q^{5}+(1+\\beta )q^{9}+2\\beta q^{11}+\\cdots" "1472.2.a.q" 2 11.753979177532473 "2.2.12.1" [] [] [0, 0, -2, -6] 1 "q+\\beta q^{3}+(-1-\\beta )q^{5}+(-3+\\beta )q^{7}+\\cdots" "1472.2.a.r" 2 11.753979177532473 "2.2.12.1" [] [] [0, 0, -2, 6] -1 "q+\\beta q^{3}+(-1+\\beta )q^{5}+(3+\\beta )q^{7}+(-3+\\cdots)q^{11}+\\cdots" "1472.2.a.s" 2 11.753979177532473 "2.2.5.1" [] [] [0, 0, 2, -2] -1 "q-\\beta q^{3}+(1-\\beta )q^{5}+(-1-\\beta )q^{7}+2q^{9}+\\cdots" "1472.2.a.t" 2 11.753979177532473 "2.2.5.1" [] [] [0, 0, 2, 2] -1 "q-\\beta q^{3}+(1+\\beta )q^{5}+(1-\\beta )q^{7}+2q^{9}+\\cdots" "1472.2.a.u" 2 11.753979177532473 "2.2.17.1" [] [] [0, 1, -4, 0] 1 "q+\\beta q^{3}-2q^{5}+(1+\\beta )q^{9}-2\\beta q^{11}+\\cdots" "1472.2.a.v" 2 11.753979177532473 "2.2.8.1" [] [] [0, 2, 4, 4] -1 "q+(1+\\beta )q^{3}+(2+\\beta )q^{5}+(2+\\beta )q^{7}+\\cdots" "1472.2.a.w" 3 11.753979177532473 "3.3.316.1" [] [] [0, -4, 2, -2] 1 "q+(-1-\\beta _{1})q^{3}+(1-\\beta _{1}+\\beta _{2})q^{5}+\\cdots" "1472.2.a.x" 3 11.753979177532473 "3.3.316.1" [] [] [0, 4, 2, 2] -1 "q+(1+\\beta _{1})q^{3}+(1-\\beta _{1}+\\beta _{2})q^{5}+(1+\\cdots)q^{7}+\\cdots" "1472.2.a.y" 4 11.753979177532473 "4.4.13768.1" [] [] [0, -2, -6, -4] 1 "q+\\beta _{2}q^{3}+(-2+\\beta _{3})q^{5}+(-1+\\beta _{1}+\\cdots)q^{7}+\\cdots" "1472.2.a.z" 4 11.753979177532473 "4.4.13768.1" [] [] [0, 2, -6, 4] -1 "q-\\beta _{2}q^{3}+(-2+\\beta _{3})q^{5}+(1-\\beta _{1})q^{7}+\\cdots" "1472.2.b.a" 6 11.753979177532473 "6.0.399424.1" [] [] [0, 0, 0, -4] NULL "q-\\beta _{3}q^{3}+(-\\beta _{3}-\\beta _{5})q^{5}+(-1+\\beta _{1}+\\cdots)q^{7}+\\cdots" "1472.2.b.b" 6 11.753979177532473 "6.0.399424.1" [] [] [0, 0, 0, 4] NULL "q-\\beta _{3}q^{3}+(\\beta _{3}+\\beta _{5})q^{5}+(1-\\beta _{1}-\\beta _{2}+\\cdots)q^{7}+\\cdots" "1472.2.b.c" 16 11.753979177532473 NULL [] [] [0, 0, 0, -4] NULL "q+\\beta _{1}q^{3}+\\beta _{11}q^{5}-\\beta _{9}q^{7}+(-1+\\beta _{2}+\\cdots)q^{9}+\\cdots" "1472.2.b.d" 16 11.753979177532473 NULL [] [] [0, 0, 0, 4] NULL "q+\\beta _{1}q^{3}-\\beta _{11}q^{5}+\\beta _{9}q^{7}+(-1+\\beta _{2}+\\cdots)q^{9}+\\cdots" "1472.2.c.a" 4 11.753979177532473 "4.0.6400.2" [] [] [0, 0, 0, 0] NULL "q+\\beta _{3}q^{3}-\\beta _{2}q^{5}-\\beta _{1}q^{7}-2q^{9}+3\\beta _{1}q^{11}+\\cdots" "1472.2.c.b" 4 11.753979177532473 "4.0.12544.2" [] [] [0, 0, 0, 0] NULL "q+\\beta _{1}q^{3}+\\beta _{2}q^{5}-\\beta _{3}q^{7}+2q^{9}-\\beta _{3}q^{11}+\\cdots" "1472.2.c.c" 6 11.753979177532473 "6.0.8869743.1" [-23] [] [0, 0, 0, 0] NULL "q-\\beta _{1}q^{3}+(-3-\\beta _{2})q^{9}+(-\\beta _{2}-\\beta _{4}+\\cdots)q^{13}+\\cdots" "1472.2.c.d" 8 11.753979177532473 "8.0.303595776.1" [] [] [0, 0, 0, 0] NULL "q-\\beta _{1}q^{3}+\\beta _{4}q^{5}-\\beta _{6}q^{7}+\\beta _{7}q^{9}+\\cdots" "1472.2.c.e" 24 11.753979177532473 NULL [] [] [0, 0, 0, 0] NULL NULL "1472.2.h.a" 8 11.753979177532473 "8.0.73358639104.3" [-184] [] [0, 0, 0, 0] NULL "q+\\beta _{3}q^{5}-3q^{9}-\\beta _{4}q^{11}+\\beta _{7}q^{19}+\\cdots" "1472.2.h.b" 8 11.753979177532473 "8.0.157351936.1" [] [] [0, 0, 0, 0] NULL "q+\\beta _{4}q^{3}+\\beta _{3}q^{5}+\\beta _{7}q^{7}+4q^{9}-\\beta _{5}q^{11}+\\cdots" "1472.2.h.c" 32 11.753979177532473 NULL [] [] [0, 0, 0, 0] NULL NULL "1472.2.i.a" 12 11.753979177532473 "12.0.322241908269256704.1" [-23] [] [0, 0, 0, 0] NULL "q+\\beta _{3}q^{3}+(-3\\beta _{1}-\\beta _{3}+\\beta _{4}+\\beta _{11})q^{9}+\\cdots" "1472.2.i.b" 80 11.753979177532473 NULL [] [] [0, 4, 0, 0] NULL NULL "1472.2.j.a" 2 11.753979177532473 "2.0.4.1" [] [] [0, -2, 0, 0] NULL "q+(i-1)q^{3}+4 i q^{7}+i q^{9}+(-4 i-4)q^{11}+\\cdots" "1472.2.j.b" 4 11.753979177532473 "4.0.144.1" [] [] [0, 6, -4, 0] NULL "q+(-\\beta_{3}-\\beta_{2}+1)q^{3}+(-\\beta_{2}-1)q^{5}+\\cdots" "1472.2.j.c" 12 11.753979177532473 "12.0.221124989353984.1" [] [] [0, -2, -4, 0] NULL "q+\\beta _{1}q^{3}+(-1+\\beta _{2}+\\beta _{3}+\\beta _{5}-\\beta _{6}+\\cdots)q^{5}+\\cdots" "1472.2.j.d" 24 11.753979177532473 NULL [] [] [0, 4, 4, 0] NULL NULL "1472.2.j.e" 46 11.753979177532473 NULL [] [] [0, -6, 4, 0] NULL NULL "1472.3.d.a" 2 40.109094913835705 "2.0.23.1" [] [] [0, 0, 12, 0] NULL "q-\\beta q^{3}+6q^{5}-14q^{9}+2\\beta q^{11}+13q^{13}+\\cdots" "1472.3.d.b" 4 40.109094913835705 "4.0.25921.1" [] [] [0, 0, -16, 0] NULL "q+(\\beta _{1}+\\beta _{2})q^{3}-4q^{5}+(-2\\beta _{1}+2\\beta _{2}+\\cdots)q^{7}+\\cdots" "1472.3.d.c" 16 40.109094913835705 NULL [] [] [0, 0, -8, 0] NULL "q+\\beta _{1}q^{3}+(-1-\\beta _{4})q^{5}+\\beta _{9}q^{7}+(-5+\\cdots)q^{9}+\\cdots" "1472.3.d.d" 20 40.109094913835705 NULL [] [] [0, 0, 8, 0] NULL "q+\\beta _{1}q^{3}+\\beta _{5}q^{5}-\\beta _{8}q^{7}+(-1+\\beta _{2}+\\cdots)q^{9}+\\cdots" "1472.3.d.e" 22 40.109094913835705 NULL [] [] [0, 0, 4, 0] NULL NULL "1472.3.d.f" 24 40.109094913835705 NULL [] [] [0, 0, 0, 0] NULL NULL "1472.3.e.a" 32 40.109094913835705 NULL [] [] [0, 0, 0, 0] NULL NULL "1472.3.e.b" 64 40.109094913835705 NULL [] [] [0, 0, 0, 0] NULL NULL "1472.3.f.a" 3 40.109094913835705 "3.3.621.1" [-23] [] [0, 0, 0, 0] NULL "q+\\beta _{1}q^{3}+(9+2\\beta _{1}+\\beta _{2})q^{9}+(-\\beta _{1}+\\cdots)q^{13}+\\cdots" "1472.3.f.b" 3 40.109094913835705 "3.3.621.1" [-23] [] [0, 0, 0, 0] NULL "q-\\beta _{1}q^{3}+(9+2\\beta _{1}+\\beta _{2})q^{9}+(-\\beta _{1}+\\cdots)q^{13}+\\cdots" "1472.3.f.c" 4 40.109094913835705 "4.0.613376.1" [] [] [0, -4, 0, 0] NULL "q+(-1+\\beta _{2})q^{3}-\\beta _{1}q^{5}+\\beta _{3}q^{7}+(-6+\\cdots)q^{9}+\\cdots" "1472.3.f.d" 4 40.109094913835705 "4.0.53792.1" [] [] [0, -2, 0, 0] NULL "q-\\beta _{3}q^{3}-\\beta _{2}q^{5}+(\\beta _{1}+\\beta _{2})q^{7}+(1+\\cdots)q^{9}+\\cdots" "1472.3.f.e" 4 40.109094913835705 "4.0.53792.1" [] [] [0, 2, 0, 0] NULL "q+\\beta _{3}q^{3}+\\beta _{2}q^{5}+(\\beta _{1}+\\beta _{2})q^{7}+(1+\\cdots)q^{9}+\\cdots" "1472.3.f.f" 4 40.109094913835705 "4.0.613376.1" [] [] [0, 4, 0, 0] NULL "q+(1-\\beta _{2})q^{3}+\\beta _{1}q^{5}+\\beta _{3}q^{7}+(-6+\\cdots)q^{9}+\\cdots" "1472.3.f.g" 12 40.109094913835705 NULL [] [] [0, 0, 0, 0] NULL "q+\\beta _{1}q^{3}-\\beta _{6}q^{5}-\\beta _{9}q^{7}+(4-\\beta _{1}+\\cdots)q^{9}+\\cdots" "1472.3.f.h" 12 40.109094913835705 NULL [] [] [0, 0, 0, 0] NULL "q-\\beta _{1}q^{3}-\\beta _{6}q^{5}+\\beta _{9}q^{7}+(4-\\beta _{1}+\\cdots)q^{9}+\\cdots" "1472.3.f.i" 24 40.109094913835705 NULL [] [] [0, 0, 0, 0] NULL NULL "1472.3.f.j" 24 40.109094913835705 NULL [] [] [0, 0, 0, 0] NULL NULL "1472.3.g.a" 32 40.109094913835705 NULL [] [] [0, 0, 0, 0] NULL NULL "1472.3.g.b" 56 40.109094913835705 NULL [] [] [0, 0, 0, 0] NULL NULL "1472.4.a.a" 1 86.85081152845018 "1.1.1.1" [] [] [0, -9, 20, -2] 1 "q-9q^{3}+20q^{5}-2q^{7}+54q^{9}-52q^{11}+\\cdots" "1472.4.a.b" 1 86.85081152845018 "1.1.1.1" [] [] [0, -8, 4, -4] 1 "q-8q^{3}+4q^{5}-4q^{7}+37q^{9}-26q^{11}+\\cdots" "1472.4.a.c" 1 86.85081152845018 "1.1.1.1" [] [] [0, -5, 6, 8] -1 "q-5q^{3}+6q^{5}+8q^{7}-2q^{9}+34q^{11}+\\cdots" "1472.4.a.d" 1 86.85081152845018 "1.1.1.1" [] [] [0, -4, -22, -8] -1 "q-4q^{3}-22q^{5}-8q^{7}-11q^{9}-20q^{11}+\\cdots" "1472.4.a.e" 1 86.85081152845018 "1.1.1.1" [] [] [0, -1, 10, 12] -1 "q-q^{3}+10q^{5}+12q^{7}-26q^{9}-42q^{11}+\\cdots" "1472.4.a.f" 1 86.85081152845018 "1.1.1.1" [] [] [0, 1, 10, -12] -1 "q+q^{3}+10q^{5}-12q^{7}-26q^{9}+42q^{11}+\\cdots" "1472.4.a.g" 1 86.85081152845018 "1.1.1.1" [] [] [0, 4, -22, 8] -1 "q+4q^{3}-22q^{5}+8q^{7}-11q^{9}+20q^{11}+\\cdots" "1472.4.a.h" 1 86.85081152845018 "1.1.1.1" [] [] [0, 5, 6, -8] -1 "q+5q^{3}+6q^{5}-8q^{7}-2q^{9}-34q^{11}+\\cdots" "1472.4.a.i" 1 86.85081152845018 "1.1.1.1" [] [] [0, 8, 4, 4] 1 "q+8q^{3}+4q^{5}+4q^{7}+37q^{9}+26q^{11}+\\cdots" "1472.4.a.j" 1 86.85081152845018 "1.1.1.1" [] [] [0, 9, 20, 2] 1 "q+9q^{3}+20q^{5}+2q^{7}+54q^{9}+52q^{11}+\\cdots" "1472.4.a.k" 2 86.85081152845018 "2.2.73.1" [] [] [0, -3, -10, 12] -1 "q+(-1-\\beta )q^{3}+(-6+2\\beta )q^{5}+(8-4\\beta )q^{7}+\\cdots" "1472.4.a.l" 2 86.85081152845018 "2.2.41.1" [] [] [0, -1, -10, -6] 1 "q+(1-3\\beta )q^{3}+(-4-2\\beta )q^{5}+(-2+\\cdots)q^{7}+\\cdots" "1472.4.a.m" 2 86.85081152845018 "2.2.41.1" [] [] [0, 1, -10, 6] 1 "q+(-1+3\\beta )q^{3}+(-4-2\\beta )q^{5}+(2+\\cdots)q^{7}+\\cdots" "1472.4.a.n" 2 86.85081152845018 "2.2.73.1" [] [] [0, 3, -10, -12] -1 "q+(1+\\beta )q^{3}+(-6+2\\beta )q^{5}+(-8+4\\beta )q^{7}+\\cdots" "1472.4.a.o" 3 86.85081152845018 "3.3.28669.1" [] [] [0, -8, 0, 42] -1 "q+(-3-\\beta _{2})q^{3}-\\beta _{1}q^{5}+(14-\\beta _{1}+\\cdots)q^{7}+\\cdots" "1472.4.a.p" 3 86.85081152845018 "3.3.1229.1" [] [] [0, -4, 10, 46] 1 "q+(-1-\\beta _{2})q^{3}+(4+\\beta _{1}-3\\beta _{2})q^{5}+\\cdots" "1472.4.a.q" 3 86.85081152845018 "3.3.761.1" [] [] [0, -3, -2, 28] -1 "q+(-1+\\beta _{2})q^{3}+(-1+\\beta _{1}+\\beta _{2})q^{5}+\\cdots" "1472.4.a.r" 3 86.85081152845018 "3.3.11032.1" [] [] [0, -2, 0, -16] 1 "q+(-1+\\beta _{1})q^{3}+(\\beta _{1}+\\beta _{2})q^{5}+(-4+\\cdots)q^{7}+\\cdots" "1472.4.a.s" 3 86.85081152845018 "3.3.761.1" [] [] [0, -1, -16, 18] 1 "q-\\beta _{1}q^{3}+(-5-2\\beta _{1}-\\beta _{2})q^{5}+(5+\\cdots)q^{7}+\\cdots" "1472.4.a.t" 3 86.85081152845018 "3.3.761.1" [] [] [0, 1, -16, -18] 1 "q+\\beta _{1}q^{3}+(-5-2\\beta _{1}-\\beta _{2})q^{5}+(-5+\\cdots)q^{7}+\\cdots" "1472.4.a.u" 3 86.85081152845018 "3.3.11032.1" [] [] [0, 2, 0, 16] -1 "q+(1-\\beta _{1})q^{3}+(\\beta _{1}+\\beta _{2})q^{5}+(4+3\\beta _{1}+\\cdots)q^{7}+\\cdots" "1472.4.a.v" 3 86.85081152845018 "3.3.761.1" [] [] [0, 3, -2, -28] -1 "q+(1-\\beta _{2})q^{3}+(-1+\\beta _{1}+\\beta _{2})q^{5}+\\cdots" "1472.4.a.w" 3 86.85081152845018 "3.3.1229.1" [] [] [0, 4, 10, -46] 1 "q+(1+\\beta _{2})q^{3}+(4+\\beta _{1}-3\\beta _{2})q^{5}+(-18+\\cdots)q^{7}+\\cdots" "1472.4.a.x" 3 86.85081152845018 "3.3.28669.1" [] [] [0, 8, 0, -42] -1 "q+(3+\\beta _{2})q^{3}-\\beta _{1}q^{5}+(-14+\\beta _{1}+\\cdots)q^{7}+\\cdots" "1472.4.a.y" 4 86.85081152845018 "4.4.334189.1" [] [] [0, -7, -14, 16] 1 "q+(-2-\\beta _{2})q^{3}+(-3-\\beta _{1}+\\beta _{3})q^{5}+\\cdots" "1472.4.a.z" 4 86.85081152845018 "4.4.2822449.1" [] [] [0, -5, 2, -32] -1 "q+(-1+\\beta _{1})q^{3}+(1+\\beta _{1}-\\beta _{3})q^{5}+\\cdots" "1472.4.a.ba" 4 86.85081152845018 "4.4.167313.1" [] [] [0, -1, 20, -10] 1 "q+\\beta _{2}q^{3}+(5-\\beta _{2}-\\beta _{3})q^{5}+(-1-\\beta _{1}+\\cdots)q^{7}+\\cdots" "1472.4.a.bb" 4 86.85081152845018 "4.4.310848.1" [] [] [0, 0, 20, -44] -1 "q+\\beta _{2}q^{3}+(5-\\beta _{1}+\\beta _{2}-\\beta _{3})q^{5}+(-11+\\cdots)q^{7}+\\cdots" "1472.4.a.bc" 4 86.85081152845018 "4.4.310848.1" [] [] [0, 0, 20, 44] 1 "q-\\beta _{2}q^{3}+(5-\\beta _{1}+\\beta _{2}-\\beta _{3})q^{5}+(11+\\cdots)q^{7}+\\cdots" "1472.4.a.bd" 4 86.85081152845018 "4.4.167313.1" [] [] [0, 1, 20, 10] 1 "q-\\beta _{2}q^{3}+(5-\\beta _{2}-\\beta _{3})q^{5}+(1+\\beta _{1}+\\cdots)q^{7}+\\cdots" "1472.4.a.be" 4 86.85081152845018 "4.4.2822449.1" [] [] [0, 5, 2, 32] -1 "q+(1-\\beta _{1})q^{3}+(1+\\beta _{1}-\\beta _{3})q^{5}+(8+\\cdots)q^{7}+\\cdots" "1472.4.a.bf" 4 86.85081152845018 "4.4.334189.1" [] [] [0, 7, -14, -16] 1 "q+(2+\\beta _{2})q^{3}+(-3-\\beta _{1}+\\beta _{3})q^{5}+\\cdots" "1472.4.a.bg" 8 86.85081152845018 NULL [] [] [0, -12, 12, -14] -1 "q+(-2+\\beta _{1})q^{3}+(1+\\beta _{3})q^{5}+(-2+\\cdots)q^{7}+\\cdots" "1472.4.a.bh" 8 86.85081152845018 NULL [] [] [0, 12, 12, 14] 1 "q+(2-\\beta _{1})q^{3}+(1+\\beta _{3})q^{5}+(2-\\beta _{7})q^{7}+\\cdots" "1472.4.a.bi" 9 86.85081152845018 NULL [] [] [0, -14, -30, -28] -1 "q+(-2+\\beta _{1})q^{3}+(-3+\\beta _{4})q^{5}+(-3+\\cdots)q^{7}+\\cdots" "1472.4.a.bj" 9 86.85081152845018 NULL [] [] [0, 0, 0, -42] -1 "q-\\beta _{2}q^{3}+\\beta _{5}q^{5}+(-5+\\beta _{3})q^{7}+(8+\\cdots)q^{9}+\\cdots" "1472.4.a.bk" 9 86.85081152845018 NULL [] [] [0, 0, 0, 42] 1 "q+\\beta _{2}q^{3}+\\beta _{5}q^{5}+(5-\\beta _{3})q^{7}+(8-\\beta _{3}+\\cdots)q^{9}+\\cdots" "1472.4.a.bl" 9 86.85081152845018 NULL [] [] [0, 14, -30, 28] 1 "q+(2-\\beta _{1})q^{3}+(-3+\\beta _{4})q^{5}+(3-\\beta _{6}+\\cdots)q^{7}+\\cdots" # Label -- # The **label** of a newform $f\in S_k^{\rm new}(N,\chi)$ has the format \( N.k.a.x \), where # - \( N\) is the level; # - \(k\) is the weight; # - \(N.a\) is the label of the Galois orbit of the Dirichlet character $\chi$; # - \(x\) is the label of the Galois orbit of the newform $f$. # For each embedding of the coefficient field of $f$ into the complex numbers, the corresponding modular form over $\C$ has a label of the form \(N.k.a.x.n.i\), where # - \(n\) determines the Conrey label \(N.n\) of the Dirichlet character \(\chi\); # - \(i\) is an integer ranging from 1 to the relative dimension of the newform that distinguishes embeddings with the same character $\chi$. # Dim -- # The **dimension** of a space of modular forms is its dimension as a complex vector space; for spaces of newforms $S_k^{\rm new}(N,\chi)$ this is the same as the dimension of the $\Q$-vector space spanned by its eigenforms. # The **dimension** of a newform refers to the dimension of its newform subspace, equivalently, the cardinality of its newform orbit. This is equal to the degree of its coefficient field (as an extension of $\Q$). # The **relative dimension** of $S_k^{\rm new}(N,\chi)$ is its dimension as a $\Q(\chi)$-vector space, where $\Q(\chi)$ is the field generated by the values of $\chi$, and similarly for newform subspaces. #$A$ (analytic_conductor) -- # The **analytic conductor** of a newform $f \in S_k^{\mathrm{new}}(N,\chi)$ is the positive real number # \[ # N\left(\frac{\exp(\psi(k/2))}{2\pi}\right)^2, # \] # where $\psi(x):=\Gamma'(x)/\Gamma(x)$ is the logarithmic derivative of the Gamma function. #Field (nf_label) -- # The **coefficient field** of a modular form is the subfield of $\C$ generated by the coefficients $a_n$ of its $q$-expansion $\sum a_nq^n$. The space of cusp forms $S_k^\mathrm{new}(N,\chi)$ has a basis of modular forms that are simultaneous eigenforms for all Hecke operators and with algebraic Fourier coefficients. For such eigenforms the coefficient field will be a number field, and Galois conjugate eigenforms will share the same coefficient field. Moreover, if $m$ is the smallest positive integer such that the values of the character $\chi$ are contained in the cyclotomic field $\Q(\zeta_m)$, the coefficient field will contain $\Q(\zeta_m)$ # For eigenforms, the coefficient field is also known as the **Hecke field**. #CM (cm_discs) -- # A newform $f$ admits a **self-twist** by a primitive # Dirichlet character $\chi$ if the equality # \[ # a_p(f) = \chi(p)a_p(f) # \] # holds for all but finitely many primes $p$. # For non-trivial $\chi$ this can hold only when $\chi$ has order $2$ and $a_p=0$ for all primes $p$ not dividing the level of $f$ for which $\chi(p)=-1$. # The character $\chi$ is then the Kronecker character of a quadratic field $K$ and may be identified by the discriminant $D$ of $K$. # If $D$ is negative, the modular form $f$ is said to have complex multiplication (CM) by $K$, and if $D$ is positive, $f$ is said to have real multiplication (RM) by $K$. The latter can occur only when $f$ is a modular form of weight $1$ whose projective image is dihedral. # It is possible for a modular form to have multiple non-trivial self twists; this occurs precisely when $f$ is a modular form of weight one whose projective image is isomorphic to $D_2:=C_2\times C_2$; in this case $f$ admits three non-trivial self twists, two of which are CM and one of which is RM. #RM (rm_discs) -- # A newform $f$ admits a **self-twist** by a primitive # Dirichlet character $\chi$ if the equality # \[ # a_p(f) = \chi(p)a_p(f) # \] # holds for all but finitely many primes $p$. # For non-trivial $\chi$ this can hold only when $\chi$ has order $2$ and $a_p=0$ for all primes $p$ not dividing the level of $f$ for which $\chi(p)=-1$. # The character $\chi$ is then the Kronecker character of a quadratic field $K$ and may be identified by the discriminant $D$ of $K$. # If $D$ is negative, the modular form $f$ is said to have complex multiplication (CM) by $K$, and if $D$ is positive, $f$ is said to have real multiplication (RM) by $K$. The latter can occur only when $f$ is a modular form of weight $1$ whose projective image is dihedral. # It is possible for a modular form to have multiple non-trivial self twists; this occurs precisely when $f$ is a modular form of weight one whose projective image is isomorphic to $D_2:=C_2\times C_2$; in this case $f$ admits three non-trivial self twists, two of which are CM and one of which is RM. #Traces (trace_display) -- # For a newform $f \in S_k^{\rm new}(\Gamma_1(N))$, its **trace form** $\mathrm{Tr}(f)$ is the sum of its distinct conjugates under $\mathrm{Aut}(\C)$ (equivalently, the sum under all embeddings of the coefficient field into $\C$). The trace form is a modular form $\mathrm{Tr}(f) \in S_k^{\rm new}(\Gamma_1(N))$ whose $q$-expansion has integral coefficients $a_n(\mathrm{Tr}(f)) \in \Z$. # The coefficient $a_1$ is equal to the dimension of the newform. # For $p$ prime, the coefficient $a_p$ is the trace of Frobenius in the direct sum of the $\ell$-adic Galois representations attached to the conjugates of $f$ (for any prime $\ell$). When $f$ has weight $k=2$, the coefficient $a_p(f)$ is the trace of Frobenius acting on the modular abelian variety associated to $f$. # For a newspace $S_k^{\rm new}(N,\chi)$, its trace form is the sum of the trace forms $\mathrm{Tr}(f)$ over all newforms $f\in S_k^{\rm new}(N,k)$; it is also a modular form in $S_k^{\rm new}(\Gamma_1(N))$. # The graphical plot displayed in the properties box on the home page of each newform or newspace is computed using the trace form. #Fricke sign (fricke_eigenval) -- # The **Fricke involution** is the Atkin-Lehner involution $w_N$ on the space $S_k(\Gamma_0(N))$ (induced by the corresponding involution on the modular curve $X_0(N)$). # For a newform $f \in S_k^{\textup{new}}(\Gamma_0(N))$, the sign of the functional equation satisfied by the L-function attached to $f$ is $i^{-k}$ times the eigenvalue of $\omega_N$ on $f$. So, for example when $k=2$, the signs swap, and the analytic rank of $f$ is even when $w_N f = -f$ and odd when $w_N f = +f$. #$q$-expansion (qexp_display) -- # The **$q$-expansion** of a modular form $f(z)$ is its Fourier expansion at the cusp $z=i\infty$, expressed as a power series $\sum_{n=0}^{\infty} a_n q^n$ in the variable $q=e^{2\pi iz}$. # For cusp forms, the constant coefficient $a_0$ of the $q$-expansion is zero. # For newforms, we have $a_1=1$ and the coefficients $a_n$ are algebraic integers in a number field $K \subseteq \C$. # Accordingly, we define the **$q$-expansion** of a newform orbit $[f]$ to be the $q$-expansion of any newform $f$ in the orbit, but with coefficients $a_n \in K$ (without an embedding into $\C$). Each embedding $K \hookrightarrow \C$ then gives rise to an embedded newform whose $q$-expansion has $a_n \in \C$, as above.